1) In class we obtained the formula
(m) m mart
for a spin 1/2 particle with magnetic moment m in a magnetic field H.
a) Develop a similar formula for a spin 1 particle (Sz = 1,0,-1) and a spin 3/2 particle
(Sz = 3/2, 1/2, -1/2, -3/2). Remember the magnetic moment is
b) In fact a general formula has been worked out for any S. The average magnetic
moment for a spin S particle is found in terms of the Brillouin function, Bs(x) according
The Brillouin function is given by
c) It is rather painful to work out this formula in general. Show that the general formula
in Eq (2) reduces to Eq (1) in the case of S = 1/2.
Hint: This last thing takes some algebra! Expand out the ctnh in terms of e+x and c and
combine the two terms in B(x). And don't give up.
2) Suppose we are not dealing with a quantum system but with a classical magnetic
moment m. (This is the case if you have a small magnetic particle that has a net magnetic
moment, but the orientation can vary)
In this case the statistical mechanics required to find in a field H (also along Z) is
not about a sum over states, but an integral over all possible angles.
a) Consider an external field H along the Z direction and use
The sum over states to find involves an integral over all angles as stated above.
Carry out this calculation and show that
< m. mL(x)
L(x) coth(x) 1/x
And x mH/kT as usual.
Now, suppose you have a completely magnetized iron sphere with a radius of 5 nm.
(Remember Fe has a Tc slightly above 1,000 K so at room temperature and below it is
pretty much fully magnetized.)
b) Estimate the total magnetic moment of the sphere remember Fe has a moment
of 2.2 18 per atom
c) If you put the sphere in a 1,000 Oe field, what is T m as a function of
temperature. Make a plot of your results from T = 0 to T = 400K. This is what
you would actually measure in an experiment.
d) Is the result in (c) similar to the result you would find if you measured a much
bigger sphere? How and why would it change?
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